How to change the background color of the Main Title in baposter?

Welcome to TeX.SE. There is no key for setting the background of the poster title. The process is:

1. Define the headerheight with headerheight=0.10textheight. This means that 10% of the height of the poster page will be used to write the poster header.
2. Set the key background=user. This means that you are going to define how the poster background will be created using the baposter macro, background{}.
3. Write the code to draw and / or fill the background. In this case, you want a different colour for the header. We can do this by writing: fill [fill=HeaderBackground] (current page.north west) rectangle ([yshift=-headerheight] current page.north east); where HeaderBackground defines your choice of colour, and we draw a rectangle equal to the width of the poster and a height equal to headerheight (as defined above).

The complete code for drawing the background is:

background{
    begin{tikzpicture}[remember picture,overlay]%
    %the poster background color (the whole page)
    fill[fill=PosterBackground] (current page.north west) rectangle 
    (current page.south east); 
    %the header (just the top of the page where the title is written)
    fill [fill=HeaderBackground] (current page.north west) rectangle 
    ([yshift=-headerheight] current page.north east); 
    end{tikzpicture}
}

Doing so produces this result:

Some other comments. headerbox is a macro that is deprecated and replaced with begin{posterbox} ... end{posterbox}. The difference is that you cannot include verbatim content in the headerbox macro, whereas because begin{posterbox} ... end{posterbox} is an environment, you can include verbatim material. If you like grid layout that can be created using baposter, you might be interested in the functionality provided by the poster library in the tcolorbox package. It was modelled on baposter, but arguably has even more functionality.

The complete code is:

PassOptionsToPackage{dvipsnames,svgnames,x11names}{xcolor} % <-- added
documentclass[landscape,archE]{baposter}

usepackage{times}
usepackage{helvet}
usepackage{palatino}
selectcolormodel{RGB}
definecolor{reddishyellow}{cmyk}{0,0.22,1.0,0.0}
definecolor{black}{cmyk}{0,0,0.0,1.0}
definecolor{DrexelBlue}{HTML}{07294D}
definecolor{Gray}{HTML}{777675}
colorlet{HeaderBackground}{NavyBlue} % <-- added
colorlet{PosterBackground}{SkyBlue}  % <-- added
newlength{leftimgwidth}

begin{document}

typeout{Poster Starts}
background{
    begin{tikzpicture}[remember picture,overlay]%
    %the poster background color
    fill[fill=PosterBackground] (current page.north west) rectangle (current page.south east); %% cover the whole poster
    %the header
    fill [fill=HeaderBackground] (current page.north west) rectangle ([yshift=-headerheight] current page.north east); % just for the poster header, fill with a different color.
    end{tikzpicture}
}

begin{poster}%
% Poster Options
{
% Show grid to help with alignment
grid=false,
columns=5,
% Column spacing
colspacing=1em,
% Color style
bgColorOne=DrexelBlue,
bgColorTwo=white,
borderColor=Gray,
headerColorOne=Gray,
headerColorTwo=reddishyellow,
headerFontColor=white,
boxColorOne=white,
%boxColorTwo=lighteryellow,
% Format of textbox
textborder=none,
textborder=rectangle,
% Format of text header
eyecatcher=false,
headerborder=open,
headershape=rectangle,
headershade=plain,
headerfont=Largesffamily, %Sans Serif  % <-- sffamily
boxshade=plain,
%background=shadeTB,                    % <-- omit
background=user, %user or none or plain % <-- added
headerheight=0.10textheight,           
linewidth=.5pt
}
% Eye Catcher
{% No eye catcher for this poster. (eyecatcher=no above).
}%If an eye catcher is present, the title is centered between eye-catcher and logo.
% Title
{sffamilycolor{white}% %Sans Serif   % <-- sffamilycolor{white}
vspace{.5em}%
textbf{Evaluating the Model}%
}
% Authors
{sffamilycolor{white}%              % <-- sffamilycolor{white}
vspace{.5em}%
John Smithtextsuperscript{1}, Harrison Franktextsuperscript{2}par%
{small%
textsuperscript{1} Doctoral Student,USApar
textsuperscript{2} USA}}
{}% Logo
begin{posterbox}[name=box1, column=0, span=1]{Header}   % <-- replace headerbox with posterbox
Some text
end{posterbox}

end{poster}

end{document}